3.162 \(\int \frac{\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx\)

Optimal. Leaf size=287 \[ -\frac{b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{8 d^4}+\frac{9 b^3 \cos \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b c}{d}+6 b x\right )}{8 d^4}+\frac{b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{8 d^4}-\frac{9 b^3 \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{8 d^4}+\frac{b^2 \sin (2 a+2 b x)}{16 d^3 (c+d x)}-\frac{3 b^2 \sin (6 a+6 b x)}{16 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3} \]

[Out]

-(b*Cos[2*a + 2*b*x])/(32*d^2*(c + d*x)^2) + (b*Cos[6*a + 6*b*x])/(32*d^2*(c + d*x)^2) - (b^3*Cos[2*a - (2*b*c
)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(8*d^4) + (9*b^3*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*c)/d + 6*b*x])/(8*
d^4) - Sin[2*a + 2*b*x]/(32*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(16*d^3*(c + d*x)) + Sin[6*a + 6*b*x]/(96*
d*(c + d*x)^3) - (3*b^2*Sin[6*a + 6*b*x])/(16*d^3*(c + d*x)) + (b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d
+ 2*b*x])/(8*d^4) - (9*b^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(8*d^4)

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Rubi [A]  time = 0.419222, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4406, 3297, 3303, 3299, 3302} \[ -\frac{b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{8 d^4}+\frac{9 b^3 \cos \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b c}{d}+6 b x\right )}{8 d^4}+\frac{b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{8 d^4}-\frac{9 b^3 \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{8 d^4}+\frac{b^2 \sin (2 a+2 b x)}{16 d^3 (c+d x)}-\frac{3 b^2 \sin (6 a+6 b x)}{16 d^3 (c+d x)}-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^4,x]

[Out]

-(b*Cos[2*a + 2*b*x])/(32*d^2*(c + d*x)^2) + (b*Cos[6*a + 6*b*x])/(32*d^2*(c + d*x)^2) - (b^3*Cos[2*a - (2*b*c
)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(8*d^4) + (9*b^3*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*c)/d + 6*b*x])/(8*
d^4) - Sin[2*a + 2*b*x]/(32*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(16*d^3*(c + d*x)) + Sin[6*a + 6*b*x]/(96*
d*(c + d*x)^3) - (3*b^2*Sin[6*a + 6*b*x])/(16*d^3*(c + d*x)) + (b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d
+ 2*b*x])/(8*d^4) - (9*b^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(8*d^4)

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx &=\int \left (\frac{3 \sin (2 a+2 b x)}{32 (c+d x)^4}-\frac{\sin (6 a+6 b x)}{32 (c+d x)^4}\right ) \, dx\\ &=-\left (\frac{1}{32} \int \frac{\sin (6 a+6 b x)}{(c+d x)^4} \, dx\right )+\frac{3}{32} \int \frac{\sin (2 a+2 b x)}{(c+d x)^4} \, dx\\ &=-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3}+\frac{b \int \frac{\cos (2 a+2 b x)}{(c+d x)^3} \, dx}{16 d}-\frac{b \int \frac{\cos (6 a+6 b x)}{(c+d x)^3} \, dx}{16 d}\\ &=-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3}-\frac{b^2 \int \frac{\sin (2 a+2 b x)}{(c+d x)^2} \, dx}{16 d^2}+\frac{\left (3 b^2\right ) \int \frac{\sin (6 a+6 b x)}{(c+d x)^2} \, dx}{16 d^2}\\ &=-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{b^2 \sin (2 a+2 b x)}{16 d^3 (c+d x)}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3}-\frac{3 b^2 \sin (6 a+6 b x)}{16 d^3 (c+d x)}-\frac{b^3 \int \frac{\cos (2 a+2 b x)}{c+d x} \, dx}{8 d^3}+\frac{\left (9 b^3\right ) \int \frac{\cos (6 a+6 b x)}{c+d x} \, dx}{8 d^3}\\ &=-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{b^2 \sin (2 a+2 b x)}{16 d^3 (c+d x)}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3}-\frac{3 b^2 \sin (6 a+6 b x)}{16 d^3 (c+d x)}+\frac{\left (9 b^3 \cos \left (6 a-\frac{6 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{6 b c}{d}+6 b x\right )}{c+d x} \, dx}{8 d^3}-\frac{\left (b^3 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{8 d^3}-\frac{\left (9 b^3 \sin \left (6 a-\frac{6 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{6 b c}{d}+6 b x\right )}{c+d x} \, dx}{8 d^3}+\frac{\left (b^3 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{8 d^3}\\ &=-\frac{b \cos (2 a+2 b x)}{32 d^2 (c+d x)^2}+\frac{b \cos (6 a+6 b x)}{32 d^2 (c+d x)^2}-\frac{b^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Ci}\left (\frac{2 b c}{d}+2 b x\right )}{8 d^4}+\frac{9 b^3 \cos \left (6 a-\frac{6 b c}{d}\right ) \text{Ci}\left (\frac{6 b c}{d}+6 b x\right )}{8 d^4}-\frac{\sin (2 a+2 b x)}{32 d (c+d x)^3}+\frac{b^2 \sin (2 a+2 b x)}{16 d^3 (c+d x)}+\frac{\sin (6 a+6 b x)}{96 d (c+d x)^3}-\frac{3 b^2 \sin (6 a+6 b x)}{16 d^3 (c+d x)}+\frac{b^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{8 d^4}-\frac{9 b^3 \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b c}{d}+6 b x\right )}{8 d^4}\\ \end{align*}

Mathematica [A]  time = 5.13101, size = 554, normalized size = 1.93 \[ \frac{12 b^3 c^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+36 b^3 c^2 d x \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )-108 b^3 c^3 \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b (c+d x)}{d}\right )-324 b^3 c^2 d x \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b (c+d x)}{d}\right )+6 b^2 c^2 d \sin (2 (a+b x))-18 b^2 c^2 d \sin (6 (a+b x))-12 b^3 (c+d x)^3 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )+108 b^3 (c+d x)^3 \cos \left (6 a-\frac{6 b c}{d}\right ) \text{CosIntegral}\left (\frac{6 b (c+d x)}{d}\right )+12 b^3 d^3 x^3 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+36 b^3 c d^2 x^2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )-108 b^3 d^3 x^3 \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b (c+d x)}{d}\right )-324 b^3 c d^2 x^2 \sin \left (6 a-\frac{6 b c}{d}\right ) \text{Si}\left (\frac{6 b (c+d x)}{d}\right )+12 b^2 c d^2 x \sin (2 (a+b x))-36 b^2 c d^2 x \sin (6 (a+b x))+6 b^2 d^3 x^2 \sin (2 (a+b x))-18 b^2 d^3 x^2 \sin (6 (a+b x))-3 b c d^2 \cos (2 (a+b x))+3 b c d^2 \cos (6 (a+b x))-3 d^3 \sin (2 (a+b x))+d^3 \sin (6 (a+b x))-3 b d^3 x \cos (2 (a+b x))+3 b d^3 x \cos (6 (a+b x))}{96 d^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^4,x]

[Out]

(-3*b*c*d^2*Cos[2*(a + b*x)] - 3*b*d^3*x*Cos[2*(a + b*x)] + 3*b*c*d^2*Cos[6*(a + b*x)] + 3*b*d^3*x*Cos[6*(a +
b*x)] - 12*b^3*(c + d*x)^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] + 108*b^3*(c + d*x)^3*Cos[6*a -
(6*b*c)/d]*CosIntegral[(6*b*(c + d*x))/d] + 6*b^2*c^2*d*Sin[2*(a + b*x)] - 3*d^3*Sin[2*(a + b*x)] + 12*b^2*c*
d^2*x*Sin[2*(a + b*x)] + 6*b^2*d^3*x^2*Sin[2*(a + b*x)] - 18*b^2*c^2*d*Sin[6*(a + b*x)] + d^3*Sin[6*(a + b*x)]
- 36*b^2*c*d^2*x*Sin[6*(a + b*x)] - 18*b^2*d^3*x^2*Sin[6*(a + b*x)] + 12*b^3*c^3*Sin[2*a - (2*b*c)/d]*SinInte
gral[(2*b*(c + d*x))/d] + 36*b^3*c^2*d*x*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 36*b^3*c*d^2*x^
2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] + 12*b^3*d^3*x^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(
c + d*x))/d] - 108*b^3*c^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d] - 324*b^3*c^2*d*x*Sin[6*a - (6*
b*c)/d]*SinIntegral[(6*b*(c + d*x))/d] - 324*b^3*c*d^2*x^2*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d]
- 108*b^3*d^3*x^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d])/(96*d^4*(c + d*x)^3)

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Maple [A]  time = 0.028, size = 404, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{{b}^{4}}{192} \left ( -2\,{\frac{\sin \left ( 6\,bx+6\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}}+2\,{\frac{1}{d} \left ( -3\,{\frac{\cos \left ( 6\,bx+6\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-3\,{\frac{1}{d} \left ( -6\,{\frac{\sin \left ( 6\,bx+6\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+6\,{\frac{1}{d} \left ( 6\,{\frac{1}{d}{\it Si} \left ( 6\,bx+6\,a+6\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 6\,{\frac{-ad+bc}{d}} \right ) }+6\,{\frac{1}{d}{\it Ci} \left ( 6\,bx+6\,a+6\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 6\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \right ) }+{\frac{3\,{b}^{4}}{64} \left ( -{\frac{2\,\sin \left ( 2\,bx+2\,a \right ) }{3\, \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{3}d}}+{\frac{2}{3\,d} \left ( -{\frac{\cos \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) ^{2}d}}-{\frac{1}{d} \left ( -2\,{\frac{\sin \left ( 2\,bx+2\,a \right ) }{ \left ( \left ( bx+a \right ) d-ad+bc \right ) d}}+2\,{\frac{1}{d} \left ( 2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 2\,{\frac{-ad+bc}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 2\,{\frac{-ad+bc}{d}} \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3*sin(b*x+a)^3/(d*x+c)^4,x)

[Out]

1/b*(-1/192*b^4*(-2*sin(6*b*x+6*a)/((b*x+a)*d-a*d+b*c)^3/d+2*(-3*cos(6*b*x+6*a)/((b*x+a)*d-a*d+b*c)^2/d-3*(-6*
sin(6*b*x+6*a)/((b*x+a)*d-a*d+b*c)/d+6*(6*Si(6*b*x+6*a+6*(-a*d+b*c)/d)*sin(6*(-a*d+b*c)/d)/d+6*Ci(6*b*x+6*a+6*
(-a*d+b*c)/d)*cos(6*(-a*d+b*c)/d)/d)/d)/d)/d)+3/64*b^4*(-2/3*sin(2*b*x+2*a)/((b*x+a)*d-a*d+b*c)^3/d+2/3*(-cos(
2*b*x+2*a)/((b*x+a)*d-a*d+b*c)^2/d-(-2*sin(2*b*x+2*a)/((b*x+a)*d-a*d+b*c)/d+2*(2*Si(2*b*x+2*a+2*(-a*d+b*c)/d)*
sin(2*(-a*d+b*c)/d)/d+2*Ci(2*b*x+2*a+2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d)/d)/d)/d))

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Maxima [C]  time = 3.22064, size = 521, normalized size = 1.82 \begin{align*} \frac{b^{4}{\left (-3 i \, E_{4}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + 3 i \, E_{4}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + b^{4}{\left (i \, E_{4}\left (\frac{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right ) - i \, E_{4}\left (-\frac{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right )\right )} \cos \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) - 3 \, b^{4}{\left (E_{4}\left (\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right ) + E_{4}\left (-\frac{2 i \, b c + 2 i \,{\left (b x + a\right )} d - 2 i \, a d}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + b^{4}{\left (E_{4}\left (\frac{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right ) + E_{4}\left (-\frac{6 i \, b c + 6 i \,{\left (b x + a\right )} d - 6 i \, a d}{d}\right )\right )} \sin \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right )}{64 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} +{\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \,{\left (b c d^{3} - a d^{4}\right )}{\left (b x + a\right )}^{2} + 3 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}{\left (b x + a\right )}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a)^3/(d*x+c)^4,x, algorithm="maxima")

[Out]

1/64*(b^4*(-3*I*exp_integral_e(4, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + 3*I*exp_integral_e(4, -(2*I*b*c +
2*I*(b*x + a)*d - 2*I*a*d)/d))*cos(-2*(b*c - a*d)/d) + b^4*(I*exp_integral_e(4, (6*I*b*c + 6*I*(b*x + a)*d -
6*I*a*d)/d) - I*exp_integral_e(4, -(6*I*b*c + 6*I*(b*x + a)*d - 6*I*a*d)/d))*cos(-6*(b*c - a*d)/d) - 3*b^4*(ex
p_integral_e(4, (2*I*b*c + 2*I*(b*x + a)*d - 2*I*a*d)/d) + exp_integral_e(4, -(2*I*b*c + 2*I*(b*x + a)*d - 2*I
*a*d)/d))*sin(-2*(b*c - a*d)/d) + b^4*(exp_integral_e(4, (6*I*b*c + 6*I*(b*x + a)*d - 6*I*a*d)/d) + exp_integr
al_e(4, -(6*I*b*c + 6*I*(b*x + a)*d - 6*I*a*d)/d))*sin(-6*(b*c - a*d)/d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^
2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d
^4)*(b*x + a))*b)

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Fricas [B]  time = 0.761782, size = 1419, normalized size = 4.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a)^3/(d*x+c)^4,x, algorithm="fricas")

[Out]

1/48*(48*(b*d^3*x + b*c*d^2)*cos(b*x + a)^6 - 72*(b*d^3*x + b*c*d^2)*cos(b*x + a)^4 + 24*(b*d^3*x + b*c*d^2)*c
os(b*x + a)^2 - 54*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*sin(-6*(b*c - a*d)/d)*sin_integra
l(6*(b*d*x + b*c)/d) + 6*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*sin(-2*(b*c - a*d)/d)*sin_i
ntegral(2*(b*d*x + b*c)/d) - 3*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(2*(b*d*
x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(-2*(b*d*x + b*c)/d))*cos(
-2*(b*c - a*d)/d) + 27*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(6*(b*d*x + b*c)
/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integral(-6*(b*d*x + b*c)/d))*cos(-6*(b*c
- a*d)/d) - 16*((18*b^2*d^3*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d - d^3)*cos(b*x + a)^5 - (18*b^2*d^3*x^2 + 36*b
^2*c*d^2*x + 18*b^2*c^2*d - d^3)*cos(b*x + a)^3 + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cos(b*x + a))*si
n(b*x + a))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**3*sin(b*x+a)**3/(d*x+c)**4,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3*sin(b*x+a)^3/(d*x+c)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError